Finite Group Databasebeta

← #16858306

Subgroups — #16858306

Order 3968.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521764C211yesmaximal, characteristic, complemented
2#14521445C211yesmaximal, characteristic, complemented
3#14521370C212yesmaximal, complemented
4#14521367C212yesmaximal, complemented
5#14521339C211yesmaximal, characteristic, complemented
6#14521326C211yesmaximal, characteristic, complemented
7#14521300C212yesmaximal, complemented
8C4×#92479C211yesmaximal, characteristic, complemented
9#14521127C211yesmaximal, characteristic, complemented
10#14521089C212yesmaximal, complemented
11C31×#547C211yesmaximal, characteristic, complemented
12C2×#41×C31—21no—
13C31×C4⋊Q8C2211yescharacteristic
14C31×#123—22no—
15C31×C22⋊Q8—21no—
16C31×#121C2212yes—
17C4×D4×C31C2211yescharacteristic
18C4×D4×C31C2211yescharacteristic
19C2×C4⋊C4×C31C2212yes—
20#11758515—21no—
21C2×#92479C2211yescharacteristic
22#11758507C2212yes—
23#11758503C2212yes—
24C2×#92474C2212yes—
25#11758469—21no—
26#11758462—22no—
27C4×D124C2211yescharacteristic
28#11758446C2211yescharacteristic
29#11758420C2211yescharacteristic
30#11758420C2212yes—
31C22⋊(C31⋊C8)C2212yes—
32#11758397C2211yescharacteristic
33#11758397C2212yes—
34#11758396C2212yes—
35#11758396C2212yes—
36#11758396C2212yes—
37#11758395C2211yescharacteristic
38#11758395C2212yes—
39C4⋊(C31⋊C8)C2211yescharacteristic
40C4×C31⋊C8C2211yescharacteristic
41#41×C31—44no—
42C2×Q8×C31—21no—
43C2×D4×C31—22no—
44C2×D4×C31D411yescharacteristic
45C2×D4×C31C2311yescharacteristic
46C22×C4×C31—22no—
47C22×C4×C31—21no—
48C22×C4×C31C2312yes—
49C2×C22⋊D31—22no—
50C2×D124C2311yescharacteristic
51C2×C4×D31—21no—
52C2×Dic62—21no—
53C4⋊C4×C31—22no—
54C4⋊C4×C31—21no—
55C4⋊C4×C31D411yescharacteristic
56C4⋊C4×C31C2311yescharacteristic
57C4⋊C4×C31C2312yes—
58C22⋊C4×C31—21no—
59C22⋊C4×C31—22no—
60C22⋊C4×C31D412yes—
61C42×C31C2311yescharacteristic
62C22⋊Dic31—22no—
63#92479—22no—
64(C2×D31)⋊C4—21no—
65(C2×D31)⋊C4—22no—
66C4⋊Dic31—21no—
67C4⋊Dic31C2311yescharacteristic
68C4⋊Dic31C2312yes—
69C4×Dic31—22no—
70#92474—24no—
71C2×C31⋊C8C2311yescharacteristic
72C2×C31⋊C8C2312yes—
73C2×C31⋊C8C2311yescharacteristic
74C23×C31—21no—
75C23×C31C2×D412yes—
76C22×D31—21no—
77Q8×C31—42no—
78D4×C31—22no—
79D4×C31—21no—
80D4×C31—21no—
81D4×C31—44no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—21no—
85C2×C4×C31—22no—
86C2×C4×C31—44no—
87C2×C4×C31—42no—
88C2×C4×C31C2×D412yes—
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2411yescharacteristic
92C22⋊D31—82no—
93C2×Dic31—21no—
94C2×Dic31—22no—
95D124—21no—
96D124—21no—
97C4×D31—42no—
98Dic62—81no—
99C31⋊C8—22no—
100C31⋊C8—21no—
101C31⋊C8#4112yescomplemented
102#3354—311nomaximal, Sylow
103C22×C31—22no—
104C22×C31—22no—
105C22×C31—22no—
106C22×C31—41no—
107C22×C31—41no—
108C22×C31—41no—
109C22×C31C22×D411yescharacteristic
110C2×D31—41no—
111C2×D31—41no—
112C2×D31—41no—
113C4×C31—22no—
114C4×C31—21no—
115C4×C31—41no—
116C4×C31—42no—
117C4×C31AGL(1,ℤ/8)12yes—
118C4×C31C2×#4111yescharacteristic
119C4×C312−1+411yescharacteristic
120Dic31—41no—
121Dic31—42no—
122#555—311no—
123#547D3111yescharacteristic, complemented
124#498—311no—
125#482—312no—
126#481—312no—
127#476—311no—
128#474—311no—
129#469—312no—
130C4×D8—311no—
131#436—311no—
132#428—312no—
133C2×C31—42no—
134C2×C31—81no—
135C2×C31#54711yescharacteristic
136C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C31#57711yescharacteristic
138D31—81no—
139C2×#41—21no—
140C2×D8—311no—
141C2×M16—312no—
142C4⋊Q8—311no—
143C4⋊Q8C2×D3111yescharacteristic
144#124—622no—
145#123—22no—
146C22⋊Q8—21no—
147#121—312no—
148#121—621no—
149#121C2×D3112yes—
150C4×D4—311no—
151C4×D4C2×D3111yescharacteristic
152C4×D4C2×D3111yescharacteristic
153#117—312no—
154#117—621no—
155C2×C4⋊C4C2×D3112yes—
156#107—311no—
157#107—312no—
158#106—312no—
159#106—312no—
160#106—312no—
161C4⋊C8—311no—
162#102—311no—
163#102—312no—
164#102—311no—
165#102—312no—
166C22⋊C8—312no—
167C4×C8—311no—
168C31#335411yescharacteristic, complemented, Sylow
169#41—44no—
170C2×Q8—21no—
171C2×Q8—621no—
172C2×D4—22no—
173C2×D4—311no—
174C2×D4—622no—
175C2×D4C22⋊D3111yescharacteristic
176C2×D4C22×D3111yescharacteristic
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—621no—
180C22×C4C22×D3112yes—
181D8—622no—
182M16—624no—
183C2×C8—311no—
184C2×C8—311no—
185C2×C8—312no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—312no—
189C4⋊C4—311no—
190C4⋊C4—621no—
191C4⋊C4C22⋊D3111yescharacteristic
192C4⋊C4C22×D3111yescharacteristic
193C4⋊C4C22×D3112yes—
194C22⋊C4—21no—
195C22⋊C4—22no—
196C22⋊C4—622no—
197C22⋊C4—622no—
198C22⋊C4—621no—
199C22⋊C4C22⋊D3112yes—
200C42—622no—
201C42C22×D3111yescharacteristic
202C23—21no—
203C23—621no—
204C23C2×C22⋊D3112yes—
205Q8—42no—
206Q8—2481no—
207D4—22no—
208D4—21no—
209D4—21no—
210D4—44no—
211D4—621no—
212D4—621no—
213D4—2482no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—621no—
221C2×C4—622no—
222C2×C4—1242no—
223C2×C4C2×C22⋊D3112yes—
224C2×C4C2×C22⋊D3111yescharacteristic
225C2×C4C2×C22⋊D3111yescharacteristic
226C2×C4C23×D3111yescharacteristic
227C8—312no—
228C8—622no—
229C8—621no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—1241no—
237C22—1241no—
238C22—1241no—
239C22C22×C22⋊D3111yescharacteristic
240C4—21no—
241C4—22no—
242C4—41no—
243C4—42no—
244C4—1242no—
245C4—1241no—
246C4#1175850912yes—
247C4#41×D3111yescharacteristic
248C4#1175856711yescharacteristic
249C2—42no—
250C2—81no—
251C2—2481no—
252C2#1452169811yescharacteristic
253C2C2×#1175850911yescharacteristic
254C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#547C2
socleC22×C31C22×D4
hypercenter#547D31
hypocenterC31#3354
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #14521764C2semidirect
2. #14521445C2semidirect
3. #14521370C2semidirect
4. #14521367C2semidirect
5. #14521339C2semidirect
6. #14521326C2semidirect
7. #14521300C2semidirect
8. C4×#92479C2semidirect
9. #14521127C2semidirect
10. #14521089C2semidirect
11. C31×#547C2semidirect
101. C31⋊C8#41semidirect
123. #547D31semidirect
168. C31#3354semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145217642
#145214452
#145213702
#145213672
#145213392
#145213262
#145213002
C4×#924792
#145211272
#145210892
C31×#5472
#335431